Customized harmonic repetitive controller and control method

ABSTRACT

The disclosure discloses a customized harmonic repetitive controller and a control method, and belongs to the field of repetitive controllers for industrial control. In the repetitive controller, a periodic signal generator formed by three time-delay modules and a positive feedforward gain module is taken as a whole to form a forward path, and an internal model of a periodic signal is constructed in the form of outputting positive feedback. Therefore, the structure of the repetitive controller conforms to a standard internal model construction method, the repetitive controller has an order expanding capability, the flexibility of the controller is greatly improved, the disturbance canceling speed of the controller is increased, and the repetitive controller is simple in structure and convenient to design. An h-order nk±m-order-harmonic repetitive controller (h≥2) obtained by further expansion covers various existing high-order repetitive controllers, and a unified form is provided to make the repetitive controller universal.

TECHNICAL FIELD

The disclosure relates to a customized harmonic repetitive controllerand a control method, and belongs to the field of repetitive controllersfor industrial control.

BACKGROUND

Repetitive control is proposed to achieve high-precision tracking ofperiodic signals. In many industrial application scenarios, it isnecessary to achieve high-precision tracking of periodic signals, suchas some production occasions where AC power provided by a grid is notdirectly used as a power source, and electric energy required to meetthe respective production occasions is obtained by various forms ofelectric energy conversion. In the electric energy conversion process,inverter equipment is required. The inverter equipment may receive thedisturbance of external signals, so an advanced control method isrequired to achieve high-precision control, and repetitive control mayachieve the purpose.

The repetitive control is simply referred to that in addition to adeviation signal, a “past control deviation”, which is a controldeviation at this moment of the previous period, is also added to aninput signal of a controlled object. The deviation from the previousrunning is reflected to the present and added to the controlled objecttogether with a “present deviation” for control. In this control mode,the deviation is repeatedly used, and is called repetitive control.After the repetitive control for several periods, the tracking precisionof a system can be greatly improved, and the quality of the system canbe improved. The control method not only is suitable for trackingperiodic input signals, but also may be used for suppressing periodicinterference. A repetitive controller is generally composed of threeparts: a repetitive signal generator internal model, a periodic delaylink and a compensator.

A traditional repetitive controller adopts a positive feedback form of adelay link (i.e. a periodic signal generator) with the delay time τ ofT₀ to construct an internal model of a periodic signal with afundamental period of T₀, and embeds the internal model into a controlloop so as to achieve static error-free tracking control or disturbancecancellation on the periodic signal (including a sinusoidal fundamentalwave and various harmonics thereof). But because the delay time of therepetitive controller from input to output is the fundamental period T₀,the response speed of the repetitive controller is relatively low. Inpractice, the repetitive controller achieves the internal model of theperiodic signal mainly in a digital mode z^(−N)(l−z^(−N)) (whereN=T₀/T_(s) is an integer and T_(s) is the sampling time), and the numberof occupied memory units is at least N₀. Therefore, the dynamic responseof the traditional repetitive controller is slow.

In order to improve the dynamic performance of the repetitivecontroller, there is a complex control strategy that combines arepetitive controller with other control methods. Although effective,this method greatly increases the difficulty and complexity ofcontroller design, and in some practical applications, harmonics thatneed to be tracked or canceled are limited to certain specificfrequencies. For example, the harmonic pollution to a power systemcaused by three-phase rectifier loads is mostly concentrated at 6k±1(k=1, 2, . . . )-order-harmonic frequencies, while the harmonicpollution to the power system caused by single-phase rectifier loads ismostly concentrated at 4k±1 (k=1, 2, . . . )-order-harmonic frequencies(i.e. odd-harmonic frequencies). In industrial occasions, the two typesof harmonics dominate. If a general repetitive controller is adopted tocancel such nk±m-order harmonics, the periodic disturbance may becanceled very slowly, so the practical requirements of the system forcontrol performance cannot be met.

In view of the need to cancel specific nk±m-order harmonics, somescholars have proposed an nk±m RC-order harmonic repetitive controller(nk±m RC), i.e. an nk±m RC proposed by Wenzhou Lu et al. in the paper “AGeneric Digital nk±m-Order Harmonic Repetitive Control Scheme for PWMConverters”, IEEE Transactions on Industrial Electronics, 2013, whichsolves the above problem. However, in practical application, if areference voltage frequency/grid voltage frequency/digital controlsystem sampling frequency changes, mismatch between a controllerinternal model and a periodic signal will occur. At this moment, if thenk±m RC proposed by Wenzhou Lu et al. is adopted, since order expansioncannot be achieved, the problem of mismatch between the controllerinternal model and the periodic signal cannot be solved, which leads tothe reduction of harmonic suppression performance, the increase ofsteady-state error, the increase of distortion, and the great decreaseof control performance.

According to the existing high-order repetitive control theory, ahigh-order repetitive controller has the ability to cope with thefrequency variation, i.e. has the ability to cope with the mismatchbetween the controller internal model and the periodic signal.Therefore, expanding the order of the controller is an effective way tosolve the problem of internal model mismatch. However, the structure ofthe nk±m RC proposed by Wenzhou Lu et al. has a forward path leadingahead of an addition loop, and does not conform to a standardconstruction structure of the high-order repetitive controller, so theorder cannot be expanded to improve the control performance, and the useof high-order nk±m RC is limited.

Therefore, it is necessary to invent a novel nk±m RC, which not only hasthe function of the traditional nk±m RC, but also is more standard instructure and may achieve order expansion to deal with the problem ofinternal model mismatch in practical application, so as to improve therobustness and flexibility of the controller.

SUMMARY

In order to solve the problem that when an existing nk±m RC faces theproblem of internal model mismatch, a high-order repetitive controllercannot be constructed to improve the controller performance, thedisclosure provides a customized harmonic repetitive controller and acontrol method.

A first objective of the disclosure is to provide a repetitivecontroller. The repetitive controller includes: a repetitive controlgain module, a positive feedforward gain module, a subtraction loop, twoaddition loops, and three identical time-delay modules.

An input end of the repetitive control gain module is used as an inputend of the repetitive controller; an output end of the repetitivecontrol gain module is used as a first input end of the first additionloop; an output end of the first addition loop is used as a first inputend of the second addition loop; an output end of the second additionloop is connected to the positive feedforward gain module and the firsttime-delay module in series respectively and then connected to apositive input end and a negative input end of the subtraction loop; anoutput end of the subtraction loop is connected to the second time-delaymodule in series and then connected to a second input end of the firstaddition loop which is also an output end of the repetitive controller;and an output end of the positive feedforward gain module is connectedto the third time-delay module in series and then connected to a secondinput end of the second addition loop.

Alternatively, the repetitive controller further includes: low passfilters and a phase lead compensation module.

The three identical time-delay modules are connected to one low passfilter in series respectively, and the output end of the subtractionloop is connected to the second time-delay module in series and thenconnected to the phase lead compensation module.

Alternatively, a transfer function of the repetitive controllerincluding the low pass filters and the phase lead compensation module isas follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi\; m}{n} )}{e^{\frac{sT_{0}}{n}} \cdot {Q(s)}}} - {Q^{2}(s)}}{e^{\frac{2sT_{0}}{n}} - {2{\cos( \frac{2\pi\; m}{n} )}{e^{\frac{sT_{0}}{n}} \cdot {Q(s)}}} + {Q^{2}(s)}} \cdot {A(s)}}}$or${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi\; m}{n} )}{z^{\frac{N}{n}} \cdot {Q(z)}}} - {Q^{2}(z)}}{z^{\frac{2N}{n}} - {2{{\cos( \frac{2\pi\; m}{n} )} \cdot z^{\frac{N}{n}} \cdot {Q(z)}}} + {Q^{2}(z)}} \cdot {A(s)}}}$

where Q(z) is the low pass filter, and A(z) is the phase leadcompensation module; c( ) is the output quantity of the repetitivecontroller; e( ) is the input quantity of the repetitive controller,i.e. the control error quantity of a control system; k_(rc) is therepetitive control gain parameter; s is the Laplace variable of acontinuous time system; and z is the variable of z transformation of adiscrete time system; N=T₀/T_(s) is an integer, T₀ is the fundamentalperiod, T₀=2π/ω₀=l/f₀, f₀ is the fundamental frequency, ω₀ is thefundamental angular frequency; T_(s) is the sampling period; n, k and mare integers not less than zero and n≠0, n>m.

When the repetitive controller adopts an analog or digital time-delaymodule respectively, expressions corresponding to c( ) are c(s) and c(z)respectively; expressions corresponding to e( ) are e(s) and e(z)respectively; expressions corresponding to Q( ) are Q(s) and Q(z)respectively; and expressions corresponding to A( ) are A(s) and A(z)respectively.

Alternatively, the low pass filter is a zero-phase low pass filter.

Alternatively, the repetitive control gain module is a proportionalityconstant for adjusting the speed of the repetitive controller to trackor cancel specific harmonics, i.e. a convergence speed of an errorbetween an output signal of the repetitive controller and a referencesignal.

A second objective of the disclosure is to provide a multi-moderepetitive controller. The multi-mode repetitive controller is formed byparallel addition of at least two of the above repetitive controllers.

A third objective of the disclosure is to provide an h-order repetitivecontroller (h≥2). The h-order repetitive controller is obtained byexpanding the above repetitive controller using the following method:accumulating Σw_(h)M^(h)( ) from 1 to h as a controller forward path,and constructing an internal model of a periodic signal in the form ofoutputting positive feedback; w_(h) is a constant coefficient, and M( )is a periodic signal generator formed by three time-delay modules and apositive feedforward gain module.

Alternatively, a transfer function of the periodic signal generator M( )of the h-order repetitive controller is as follows:

${M(s)} = {\frac{e^{- \frac{sT_{0}}{n}}\lbrack {{\cos( \frac{2\pi m}{n} )} - e^{\frac{sT_{0}}{n}}} \rbrack}{1 - {e^{- \frac{{sT}_{0}}{n}}{\cos( \frac{2\pi m}{n} )}}} = \frac{{e^{\frac{sT_{0}}{n}}{\cos( \frac{2\pi m}{n} )}} - 1}{e^{\frac{2sT_{0}}{n}} - {e^{\frac{sT_{0}}{n}}{\cos( \frac{2\pi m}{n} )}}}}$or${M(z)} = {\frac{z^{- \frac{N}{n}}\lbrack {{\cos( \frac{2\pi m}{n} )} - z^{- \frac{N}{n}}} \rbrack}{1 - {z^{- \frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}} = \frac{{z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}} - 1}{z^{\frac{2N}{n}} - {z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}}}$

M(s) is the transfer function of the periodic signal generator M( ) ofthe h-order repetitive controller when the repetitive controller adoptsan analog time-delay module, and M(z) is the transfer function of theperiodic signal generator M( ) of the h-order repetitive controller whenthe repetitive controller adopts a digital time-delay module.

A fourth objective of the disclosure is to provide a converter. Theconverter is controlled by the above repetitive controller, or the abovemulti-mode repetitive controller, or the above h-order repetitivecontroller. The converter includes an inverter and a rectifier.

A fifth objective of the disclosure is to provide a control method of arepetitive controller. The method is used for canceling nk±m-orderharmonics using the above repetitive controller or the above multi-moderepetitive controller or the above h-order repetitive controller; n, kand m are integers not less than zero and n≠0, n>m. The method includes:

using a repetitive control gain module to perform repetitive controlgain on an input quantity of the repetitive controller to obtain anoutput quantity of the repetitive control gain module;

using a positive feedforward gain module to perform positive feedforwardgain on an output quantity of a second addition loop to obtain an outputquantity of the positive feedforward gain module;

using a first addition loop to add the output quantity of the repetitivecontrol gain module and an output quantity of a subtraction loop outputby a second time-delay module in a delay manner to obtain an outputquantity of the first addition loop;

using the second addition loop to add the output quantity of the firstaddition loop and the output quantity of the positive feedforward gainmodule output by a third time-delay module in a delay manner to obtainan output quantity of the second addition loop;

using the subtraction loop to subtract the output quantity of thepositive feedforward gain module from the output quantity of the secondaddition loop output by a first time-delay module in a delay manner toobtain the output quantity of the subtraction loop;

using the first time-delay module to output the output quantity of thesecond addition loop in a delay manner;

using the second time-delay module to output the output quantity of thesubtraction loop in a delay manner; and

using the third time-delay module to output the output quantity of thepositive feedforward gain module in a delay manner.

Alternatively, the method further includes:

adjusting the repetitive control gain to adjust the speed of therepetitive controller to track or cancel specific harmonics, i.e. aconvergence speed of an error between an output signal of the repetitivecontroller and a reference signal.

Alternatively, the method further includes:

determining parameters of the positive feedforward gain module accordingto the order of harmonics to be tracked or canceled.

Alternatively, the time-delay module is an analog or digital time-delaymodule, and a transfer function of the repetitive controller is asfollows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}} - 1}{e^{\frac{2sT_{0}}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}}} + 1}}}$or${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot z^{\frac{N}{n}}} - 1}{z^{\frac{2N}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot z^{\frac{N}{n}}}} + 1}}}$

c( ) is the output quantity of the repetitive controller; e( ) is theinput quantity of the repetitive controller, i.e. the control errorquantity of a control system; k_(rc) is the repetitive control gainparameter; s is the Laplace variable of a continuous time system; and zis the variable of z transformation of a discrete time system;N=T₀/T_(s) is an integer, T₀ is the fundamental period, T₀=2π/ω₀=l/f₀,f₀ is the fundamental frequency, ω₀ is the fundamental angularfrequency, T_(s) is the sampling period; n, k and m are integers notless than zero and n≠0, n>m.

When the repetitive controller adopts an analog or digital time-delaymodule respectively, expressions corresponding to c( ) are c(s) and c(z)respectively; and expressions corresponding to e( ) are e(s) and e(z)respectively.

Alternatively, the method further includes: adding the above repetitivecontroller to a feedback control system in an insertion manner forcanceling nk±m-order harmonics components in control errors; when therepetitive controller is added to the feedback control system in theinsertion manner, the transfer function of the repetitive controller is:

${G_{rc}(z)} = {{\frac{k_{rc}}{2}\lbrack {\frac{e^{j\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}{1 - {e^{j\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}} + \frac{e^{{- j}\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}{1 - {e^{{- j}\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}}} \rbrack} \cdot {A(z)}}$

Q(z) is a low pass filter, and A(z) is a phase lead compensation module.

The feedback control system is stable when the following two conditionsare met:

(1) poles of a transfer function of a closed-loop system before therepetitive controller is inserted are located in a unit circle; and

(2) the repetitive control gain parameter k_(rc) in the insertedcontroller meets 0<k_(rc)<2.

The disclosure has the following beneficial effects:

1. The structure of the repetitive controller provided by the disclosureconforms to a standard internal model construction method, i.e. theperiodic signal generator formed by three time-delay modules and thepositive feedforward gain module is taken as a whole to form the forwardpath, and the internal model of the periodic signal is constructed inthe form of outputting positive feedback. According to the existinghigh-order repetitive control theory, a high-order repetitive controllermay be formed only when the periodic signal generator is used as aunique forward path as a whole, and the internal model is constructed inthe form of outputting positive feedback. In the repetitive controllerprovided by the disclosure, the periodic signal generator is used as theunique forward path as a whole, and the internal model is constructed inthe form of outputting positive feedback, so that the repetitivecontroller has an order expanding capability, and the flexibility of thecontroller is greatly improved.

2. The h-order nk±m RC (h≥2) obtained by further expanding therepetitive controller provided by the disclosure covers various existinghigh-order repetitive controllers, and a unified form is provided. Forexample, a high-order basic repetitive controller applied in Dapeng Liet al. “Second-order RC: analysis, augmentation, andanti-frequency-variation for single-phase grid-tied inverter”, IET PowerElectronics, 2018, is a special case of the h-order nk±m RC of thedisclosure when h=2, n=1, m=0. For another example, a high-orderodd-harmonic repetitive controller applied in Ramos G A et al. “Powerfactor correction and harmonic compensation using second-orderodd-harmonic repetitive control”, IET control theory & applications, isa special case of the h-order nk±m RC of the disclosure when h=2, n=4,m=1. Therefore, the h-order nk±m RC provided by the disclosure hasuniversality.

3. The h-order nk±m RC (h≥2) obtained by further expanding therepetitive controller provided by the disclosure has a higher errorconvergence speed, and has a certain suppression effect when an internalmodel and a periodic signal are mismatched near a resonance frequencypoint, so that the performance of the controller is improved.

4. The repetitive controller provided by the disclosure is speciallyused for carrying out error-free tracking or disturbance cancellation onnk±m-order-harmonic signals, and different n and m values may becustomized according to actual requirements for canceling harmonicdisturbance signals or tracking reference signals. For the requirementsof canceling (6k±1)-order harmonics and tracking a fundamental referencesignal in three-phase inversion, only n=6 and m=1 are needed. For therequirements of canceling odd-harmonics and tracking a fundamentalreference signal in single-phase inversion, only n=4 and m=1 are needed.Moreover, three delay links in the repetitive controller provided by thedisclosure are completely identical, the delay time τ of the delay linksis equal to 1/n times of the fundamental period T₀, and the longestdelay time path is composed of two of the delay links, so that the totaldelay time is (2T₀/n)<T₀. Therefore, the response speed of therepetitive controller provided by the disclosure is much higher thanthat of a general repetitive controller under the condition that therepetitive control gain k_(rc) is the same, and the speed of disturbancecancellation is greatly increased.

5. The multi-mode repetitive controller provided by the disclosure maybe used for canceling all harmonics or any harmonic and mayindependently adjust the control gain of each harmonic controller.

6. The repetitive controller provided by the disclosure only needs onepositive feedforward coefficient module besides the three time-delaymodules and the repetitive control gain module, so that the controlleris simple in structure and convenient to design.

7. When the repetitive controller provided by the disclosure is used forcanceling the disturbance that the ratio of two frequencies of nk±m andnk−m is not an integer multiple relationship, only one time-delay linkis needed to construct a disturbance signal internal model, so that thedesign of the time-delay link in the repetitive controller issimplified.

8. The three delay links in the repetitive controller provided by thedisclosure are completely identical, and the number of occupied memoryunits is N/n, so that the total number of memory units of the repetitivecontroller is (3N/n), and the number of occupied memory units of annk±m-order-harmonic digital repetitive controller is greatly lower thanthat of a general digital repetitive controller.

BRIEF DESCRIPTION OF FIGURES

In order to more clearly illustrate the technical solutions of theexamples of the disclosure, the drawings used in the description of theexamples are briefly described below, and it is obvious that thedrawings in the description below are only some examples of thedisclosure, and a person of ordinary skill in the art can obtain otherdrawings from these drawings without any creative effort.

FIG. 1 is a structural block diagram of an nk±m RC according to thedisclosure.

FIG. 2 is a digital implementation form of FIG. 1, and is a structuralblock diagram of an nk±m-order-harmonic digital repetitive controller.

FIG. 3 is a structural block diagram of a periodic signal generator in aforward path in a digital form of an nk±m RC according to thedisclosure.

FIG. 4 is a structural block diagram of an improved nk±m RC added with alow pass filter link and a phase lead compensation link on the basis ofFIG. 1.

FIG. 5 is a digital implementation form of FIG. 4, and is a structuralblock diagram of an improved nk±m-order-harmonic digital repetitivecontroller.

FIG. 6 is a structural block diagram of an h-order nk±m-order-harmonicdigital repetitive controller (h≥2) expanded on the basis of FIG. 2.

FIG. 7 is a structural block diagram of an improved h-ordernk±m-order-harmonic digital repetitive controller (h≥2).

FIG. 8 is a structural block diagram of an improved h-ordernk±m-order-harmonic digital repetitive controller when h=2.

FIG. 9 is a structural block diagram of an all-harmonic repetitivecontroller with a parallel structure formed by parallel addition of nk±mRC according to the disclosure, where the repetitive controller withthis structure may cancel all harmonics.

FIG. 10 is a structural block diagram of an any-harmonic repetitivecontroller with a parallel structure formed by parallel addition of nk±mRC according to the disclosure, where the repetitive controller withthis structure may cancel any harmonic.

FIG. 11 is a digital implementation form of FIG. 9, and is a structuralblock diagram of an all-harmonic digital repetitive controller with aparallel structure.

FIG. 12 is a digital implementation form of FIG. 10, and is a structuralblock diagram of an any-harmonic digital repetitive controller with aparallel structure.

FIG. 13 is a structural block diagram of an improved all-harmonicrepetitive controller with a parallel structure added with a low passfilter link and a phase lead compensation link on the basis of FIG. 9.

FIG. 14 is a structural block diagram of an improved any-harmonicrepetitive controller with a parallel structure added with a low passfilter link and a phase lead compensation link on the basis of FIG. 10.

FIG. 15 is a digital implementation form of FIG. 13, and is a structuralblock diagram of an improved all-harmonic digital repetitive controllerwith a parallel structure.

FIG. 16 is a digital implementation form of FIG. 14, and is a structuralblock diagram of an improved any-harmonic digital repetitive controllerwith a parallel structure.

FIG. 17 is a structural block diagram of a control system in which animproved nk±m-order-harmonic digital repetitive controller superposedwith a general feedback controller.

FIG. 18 is a structural block diagram of a control system in which animproved h-order nk±m-order-harmonic digital repetitive controller (h≥2)superposed with a general feedback controller.

FIG. 19A is a steady-state output oscillogram under two complex controlsin application of an nk±m RC (n=1 and m=0) according to the disclosurein combination with a general feedback controller.

FIG. 19B is an error convergence variation diagram under two complexcontrols in application of an nk±m RC (n=1 and m=0) according to thedisclosure in combination with a general feedback controller.

FIG. 19C is a steady-state output oscillogram under two complex controlsin application of an nk±m RC (n=4 and m=1) according to the disclosurein combination with a general feedback controller.

FIG. 19D is an error convergence variation diagram under two complexcontrols in application of an nk±m RC (n=4 and m=1) according to thedisclosure in combination with a general feedback controller.

FIG. 20A is a steady-state output oscillogram under two complex controlsin application of an nk±m RC (n=1 and m=0) according to the disclosurein combination with a general feedback controller.

FIG. 20B is an error convergence variation diagram under two complexcontrols in application of an nk±m RC (n=1 and m=0) according to thedisclosure in combination with a general feedback controller.

FIG. 20C is a steady-state output oscillogram under two complex controlsin application of an nk±m RC (n=6 and m=1) according to the disclosurein combination with a general feedback controller.

FIG. 20D is an error convergence variation diagram under two complexcontrols in application of an nk±m RC (n=6 and m=1) according to thedisclosure in combination with a general feedback controller.

FIG. 21A is an error convergence variation diagram of a controller whenthe reference voltage frequency is changed from 50 Hz to 49.8 Hz, i.e.an internal model of the controller is mismatched with a periodic signalto be tracked or canceled, taking 6k±1 RC as an example.

FIG. 21B is a harmonic spectrogram of a controller when the referencevoltage frequency is changed from 50 Hz to 49.8 Hz, i.e. an internalmodel of the controller is mismatched with a periodic signal to betracked or canceled, taking 6k±1 RC as an example.

FIG. 21C is an error convergence variation diagram of a controller whenthe reference voltage frequency is changed from 50 Hz to 49.8 Hz, i.e.an internal model of the controller is mismatched with a periodic signalto be tracked or canceled, taking second-order 6k±1 RC as an example.

FIG. 21D is a harmonic spectrogram of a controller when the referencevoltage frequency is changed from 50 Hz to 49.8 Hz, i.e. an internalmodel of the controller is mismatched with a periodic signal to betracked or canceled, taking second-order 6k±1 RC as an example.

FIG. 22 is a control block diagram of adding an nk±m-order-harmonicdigital repetitive controller or a multi-mode digital repetitivecontroller or an h-order nk±m-order-harmonic digital repetitivecontroller according to the disclosure to a control system in a cascademanner.

DETAILED DESCRIPTION

In order to make objectives, technical solutions, and advantages of thedisclosure more apparent, implementations of the disclosure will bedescribed in further detail with reference to the drawings.

Example 1

This example provides a repetitive controller. The repetitive controllerincludes: a repetitive control gain module, a positive feedforward gainmodule, a subtraction loop, two addition loops, and three identicaltime-delay modules.

An input end of the repetitive control gain module is used as an inputend of the repetitive controller; an output end of the repetitivecontrol gain module is used as a first input end of the first additionloop; an output end of the first addition loop is used as a first inputend of the second addition loop; an output end of the second additionloop is connected to the positive feedforward gain module and the firsttime-delay module in series respectively and then connected to apositive input end and a negative input end of the subtraction loop; anoutput end of the subtraction loop is connected to the second time-delaymodule in series and then connected to a second input end of the firstaddition loop, which is also an output end of the repetitive controller;and an output end of the positive feedforward gain module is connectedto the third time-delay module in series and then connected to a secondinput end of the second addition loop.

FIG. 1 shows a structural block diagram of a customized harmonicrepetitive controller provided by the disclosure. c(s) is the outputquantity of the repetitive controller; e(s) is the input quantity of therepetitive controller, i.e. the control error quantity of a controlsystem; and k_(rc) is the repetitive control gain module.

A transfer function of the customized harmonic repetitive controller isas follows:

$\begin{matrix}{{G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}} - 1}{e^{\frac{2sT_{0}}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}}} + 1}}}} & (1)\end{matrix}$

In formula (1), k_(rc) is the repetitive control gain parameter; T₀ isthe fundamental period, T₀=2π/Ω₀=l/f₀, f₀ is the fundamental frequency,ω₀ is the fundamental angular frequency; n, k and m are integers notless than zero and n≠0, n>m; and e is a natural constant.

By adjusting the value of the gain coefficient k_(rc), the errorconvergence speed of the system may be changed; the larger k_(rc) is,the higher the steady-state error convergence speed of the system is,but the system will be out of a stable range due to over-large k_(rc),so k_(rc) can increase the convergence speed of the system only within acertain range.

Three delay links in FIG. 1 are completely identical, the delay time τof the delay links is equal to 1/n times of the fundamental period T₀,and the longest delay time path is composed of two of the delay links,so that the total delay time is (2T₀/n)<T₀. Therefore, the responsespeed of the customized harmonic repetitive controller provided by thedisclosure is much higher than that of a general repetitive controllerunder the condition that the repetitive control gain k_(rc) is the same.This is a great advantage of the nk±m RC, and besides the threeidentical time-delay links and the repetitive control gain module, onlyone positive feedforward gain module exists in the controller, so thatthe controller is simple in structure and more convenient to design.

Formula (1) is transformed as follows:

${G_{rc}(s)} = {{k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}} - 1}{e^{\frac{2sT_{0}}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}}} + 1}} = {{k_{rc} \cdot \frac{{\cos( \frac{2\pi m}{n} )} - e^{- \frac{sT_{0}}{n}}}{e^{\frac{sT_{0}}{n}} + e^{- \frac{sT_{0}}{n}} - {2{\cos( \frac{2\pi m}{n} )}}}} = {{\frac{1}{2}{k_{rc} \cdot \frac{{\cos( \frac{2\pi m}{n} )} - e^{- \frac{sT_{0}}{n}}}{{\cosh( \frac{sT_{0}}{n} )} - {\cos( \frac{2\pi m}{n} )}}}} = {\frac{1}{2}{k_{rc} \cdot \frac{{\cos( \frac{2\pi m}{n} )} - e^{- \frac{sT_{0}}{n}}}{\begin{matrix}{2{\sin^{2}( \frac{m\pi}{n} )}{( {1 + \frac{s^{2}}{m^{2}\omega_{0}^{2}}} ) \cdot}} \\{\prod\limits_{k = 1}^{\infty}\{ {\lbrack {1 + \frac{s^{2}}{( {{nk} + m} )^{2}\omega_{0}^{2}}} \rbrack\lbrack {1 + \frac{s^{2}}{( {{nk} - m} )^{2}\omega_{0}^{2}}} \rbrack} \}}\end{matrix}}}}}}}$

The above formula requires m≠0.

When m=0, the transfer function of the repetitive controller forcanceling nk±m-order harmonics may be formulated as follows:

${{{G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}} - 1}{e^{\frac{2sT_{0}}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot e^{\frac{sT_{0}}{n}}}} + 1}}}}}_{m = 0} = {{k_{rc} \cdot \frac{e^{\frac{sT_{0}}{n}} - 1}{e^{\frac{2sT_{0}}{n}} - {2e^{\frac{sT_{0}}{n}}} + 1}} = {{k_{rc} \cdot \frac{e^{\frac{sT_{0}}{n}} - 1}{( {e^{\frac{sT_{0}}{n}} - 1} )^{2}}} = {{k_{rc} \cdot \frac{1}{e^{\frac{sT_{0}}{n}} - 1}} = {k_{rc} \cdot \frac{1}{( {s{T_{0}/n}} ) \cdot e^{\frac{sT_{0}}{2n}} \cdot {\prod\limits_{k = 1}^{\infty}\lbrack {1 + {s^{2}/( {{nk}\;\omega_{0}} )^{2}}} \rbrack}}}}}}$

By combining the above two formulae, it is thus obtained that a pole ofthe repetitive controller shown in FIG. 1 is at the frequency of(nk±m)ω₀, i.e. the pole frequency is mω0, (n±m)ω₀, (2n±m)ω₀, (in±m)ω₀, .. . , (where i=1, 2, 3 . . . ).

Since the gain of the repetitive controller is infinite at the frequencyof (nk±m)ω₀, harmonic components with the frequency of (nk±m)ω₀ in thecontrol error e(s) can be completely canceled, so that completecancellation or error-free tracking of nk±m-order-harmonic disturbanceis realized. Therefore, the repetitive controller, i.e. the customizedharmonic repetitive controller provided by the disclosure is called annk±m RC.

In practical application, m and n may be endowed with different valuesaccording to requirements of different occasions, so that error-freetracking or disturbance suppression of specific nk±m-order harmonics maybe realized. For example, in the case of a three-phase inverter withthree-phase rectifier loads, since harmonics are mainly concentrated at(6k±1) (i.e. 5, 7, 11, 13 and the like)-order-harmonic frequencycomponents and it is often necessary to track a fundamental referencesignal, it only needs to make n=6 and m=1, and error-free tracking ofthe fundamental reference signal and complete cancellation of(6k±1)-order-harmonics may be realized. In the case of a single-phaseinverter with single-phase rectifier loads, since harmonics are mainlyconcentrated at (4k±1) (i.e. odd orders of 3, 5, 7, 9 and thelike)-frequency components and it is often necessary to track afundamental reference signal, it only needs to make n=4 and m=1, anderror-free tracking of the fundamental reference signal and completecancellation of odd-harmonics may be realized.

In practice, the repetitive controller is mostly implemented and applieddigitally. The digital implementation corresponding to the repetitivecontroller shown in FIG. 1 is as shown in FIG. 2 with a transferfunction as follows:

${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )} \cdot z^{\frac{N}{n}}} - 1}{z^{\frac{2N}{n}} - {2{{\cos( \frac{2\pi m}{n} )} \cdot z^{\frac{N}{n}}}} + 1}}}$

c(z) is the output quantity of the repetitive controller; e(z) is theinput quantity of the repetitive controller, i.e. the control errorquantity of a control system; k_(rc) is the repetitive control gain;N=T₀/T_(s) is an integer, T₀ is the fundamental period, T₀=2T/ω₀=l/f₀,f₀ is the fundamental frequency, ω₀ is the fundamental angularfrequency, T_(s) is the sampling period; n, k and m are integers notless than zero and n≠0, n>m.

Three time-delay links in FIG. 2 are completely identical, and thenumber of occupied memory units is N/n, so that the total number ofmemory units of the repetitive controller is (3N/n). Therefore, thenk±m-order-harmonic digital repetitive controller occupies much lessmemory space than a general digital repetitive controller.

FIG. 3 shows a digital form of a periodic signal generator in a forwardpath in an nk±m RC according to the disclosure, which is composed ofthree completely-identical time-delay modules and a positive feedforwardgain module. A transfer function M(z) may be expressed as follows:

${M(z)} = {\frac{z^{- \frac{N}{n}}\lbrack {{\cos( \frac{2\pi m}{n} )} - z^{- \frac{N}{n}}} \rbrack}{1 - {z^{- \frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}} = \frac{{z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}} - 1}{z^{\frac{2N}{n}} - {z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}}}$

In practical application, in order to improve the stability andanti-interference capability of the control system, it is usuallynecessary to improve the nk±m RC in FIG. 1 or FIG. 2 by adding a lowpass filter link Q(s) or Q(z) and a phase lead compensation link A(s) orA(z) to the repetitive controller, as shown in FIG. 4 and FIG. 5, whereFIG. 5 is a digital implementation form of FIG. 4.

The transfer function of the improved nk±m RC shown in FIG. 4 may bewritten as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi m}{n} )}{e^{\frac{sT_{0}}{n}} \cdot {Q(s)}}} - {Q^{2}(s)}}{e^{\frac{2sT_{0}}{n}} - {2{\cos( \frac{2\pi m}{n} )}{e^{\frac{sT_{0}}{n}} \cdot {Q(s)}}} + {Q^{2}(s)}} \cdot {A(s)}}}$

The transfer function of the improved nk±m-order-harmonic digitalrepetitive controller shown in FIG. 5 may be written as follows:

$\begin{matrix}{{G_{rc}(z)} = {\frac{c(z)}{e(z)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi\; m}{n} )}{z^{\frac{2N}{n}} \cdot {Q(z)}}} - {Q^{2}(z)}}{z^{\frac{2N}{n}} - {2{{\cos( \frac{2\pi\; m}{n} )} \cdot z^{\frac{2N}{n}} \cdot {Q(z)}}} + {Q^{2}(z)}} \cdot {A(z)}}}} & \;\end{matrix}$

FIG. 6 shows a universal structural block diagram of an h-ordernk±m-order-harmonic digital repetitive controller (h≥2). In practicalapplication, the h-order nk±m RC is also implemented digitally, and thetransfer function may be written as follows:

${G_{{hO} - {rc}}(z)} = {k_{rc} \cdot \frac{\sum\limits_{h = 1}^{h}{w_{h}{M^{h}(z)}}}{1 - {\sum\limits_{h = 1}^{h}{w_{h}{M^{h}(z)}}}}}$${{where}\mspace{14mu}{M(z)}} = {\frac{z^{- \frac{N}{n}}\lbrack {{\cos( \frac{2\pi\; m}{n} )} - z^{- \frac{N}{n}}} \rbrack}{1 - {z^{- \frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}}} = \frac{{z^{\frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}} - 1}{z^{\frac{N}{n}} - {z^{\frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}}}}$

FIG. 7 shows an improved h-order nk±m-order-harmonic digital repetitivecontroller (h≥2). Different from a first-order nk±m-order-harmonicdigital repetitive controller, due to the increase of the order, ifoutput ends of the three identical time-delay modules are respectivelyconnected to a low pass filter Q(z) in series for filtering, thecomplexity of the controller is inevitably greatly increased, and thedesign difficulty is also increased. Therefore, for the h-ordernk±m-order-harmonic digital repetitive controller (h≥2) provided by thedisclosure, the low pass filter Q(z) is uniformly connected in seriesbehind

${\sum\limits_{h = 1}^{h}{w_{h}{M^{h}(z)}}},$and a phase lead compensation module A(z) is still connected in seriesto the output end of the repetitive controller. The transfer function ofthe improved h-order nk±m-order-harmonic digital repetitive controller(h≥2) may be written as follows:

${G_{{hO} - {rc}}(z)} = {k_{rc} \cdot \frac{{A(z)} \cdot {Q(z)} \cdot {\sum\limits_{h = 1}^{h}{w_{h}{M^{h}(z)}}}}{1 - {{Q(z)} \cdot {\sum\limits_{h = 1}^{h}{w_{h}{M^{h}(z)}}}}}}$${{where}\mspace{14mu}{M(z)}} = {\frac{z^{- \frac{N}{n}}\lbrack {{\cos( \frac{2\pi\; m}{n} )} - z^{- \frac{N}{n}}} \rbrack}{1 - {z^{- \frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}}} = \frac{{z^{\frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}} - 1}{z^{\frac{N}{n}} - {z^{\frac{N}{n}}{\cos( \frac{2\pi\; m}{n} )}}}}$

FIG. 8 shows an example of an h-order nk±m-order-harmonic digitalrepetitive controller when h=2. In order to avoid increasing thecomplexity and design difficulty of the controller due to the increaseof the order, second-order, i.e. a second-order nk±m-order-harmonicdigital repetitive controller is enough. The transfer function is asfollows:

${G_{{SO} - {rc}}(z)} = {k_{rc} \cdot \frac{{M(z)}\lbrack {w_{1} + {w_{2}{M(z)}}} \rbrack}{1 - {{M(z)}\lbrack {w_{1} + {w_{2}{M(z)}}} \rbrack}}}$${{where}\mspace{14mu}{M(z)}} = {\frac{z^{\frac{N}{n}}\lbrack {{\cos( \frac{2\pi m}{n} )} - z^{\frac{N}{n}}} \rbrack}{1 - {z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}} = \frac{{z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}} - 1}{z^{\frac{2N}{n}} - {z^{\frac{N}{n}}{\cos( \frac{2\pi m}{n} )}}}}$

FIG. 9 shows an all-harmonic repetitive controller with a parallelstructure formed by parallel addition of nk±m RCs according to thedisclosure. For different values of n, m and k, the repetitivecontroller with this structure may cancel all harmonics, and mayindependently adjust the control gain of each harmonic. The transferfunction is as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {\sum\limits_{i = 0}^{m}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n}}} - 1}{e^{\frac{2{sT}_{0}}{n}} - {2{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n}}}} + 1}}}}$

n and m are both positive integers; when n is an even number, m=0, 1, .. . , n/2; and when n is an odd number, m=0, 1, . . . , [n/2].

FIG. 10 shows an any-harmonic repetitive controller with a parallelstructure formed by parallel addition of nk±m RCs according to thedisclosure. For different values of n, m and k, the repetitivecontroller with this structure may cancel any harmonic, and mayindependently adjust the control gain of each harmonic. The transferfunction is as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {\sum\limits_{i = 0}^{l}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n_{i}}}} - 1}{e^{\frac{2{sT}_{0}}{n_{i}}} - {2{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n_{i}}}}} + 1}}}}$

n_(i) and m_(i) are any positive integers.

FIG. 11 shows a digital form of an all-harmonic repetitive controllerwith a parallel structure formed by parallel addition of nk±m RCaccording to the disclosure. The transfer function is as follows:

${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {\sum\limits_{i = 0}^{m}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot z^{\frac{N}{n}}} - 1}{z^{\frac{2N}{n}} - {2{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot z^{\frac{N}{n}}}} + 1}}}}$

n and m are both positive integers; when n is an even number, m=0, 1, .. . , n/2; and when n is an odd number, m=0, 1, . . . , [n/2].

FIG. 12 shows a digital form of an any-harmonic repetitive controllerwith a parallel structure formed by parallel addition of nk±m RCaccording to the disclosure. The transfer function is as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {\sum\limits_{i = 0}^{l}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n_{i}} \cdot m_{i}} )} \cdot z^{\frac{N}{n_{i}}}} - 1}{z^{\frac{2N}{n_{i}}} - {2{{\cos( {\frac{2\pi}{n_{i}} \cdot m_{i}} )} \cdot z^{\frac{N}{n_{i}}}}} + 1}}}}$

n_(i) and m_(i) are any positive integers.

FIG. 13 shows a structural block diagram of an improved all-harmonicrepetitive controller with a parallel structure added with a low passfilter link Q(s) and a phase lead compensation link A(s). The transferfunction is as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {\sum\limits_{i = 0}^{l}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n_{i}} \cdot m_{i}} )} \cdot e^{\frac{{sT}_{0}}{n_{i}}} \cdot {Q_{i}(s)}} - {Q_{i}^{2}(s)}}{e^{\frac{2{sT}_{0}}{n_{i}}} - {2{{\cos( {\frac{2\pi}{n_{i}} \cdot m_{i}} )} \cdot e^{\frac{{sT}_{0}}{n_{i}}} \cdot {Q_{i}(s)}}} - {Q_{i}^{2}(s)}} \cdot {A(s)}}}}$

n and m are both positive integers; when n is an even number, m=0, 1, .. . , n/2; and when n is an odd number, m=0, 1, . . . , [n/2].

FIG. 14 shows a structural block diagram of an improved any-harmonicrepetitive controller with a parallel structure added with a low passfilter link Q(s) and a phase lead compensation link A(s). The transferfunction is as follows:

${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {\sum\limits_{i = 0}^{m}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n}} \cdot {Q_{i}(s)}} - {Q_{i}^{2}(s)}}{e^{\frac{2{sT}_{0}}{n}} - {2{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot e^{\frac{{sT}_{0}}{n}} \cdot {Q_{i}(s)}}} - {Q_{i}^{2}(s)}} \cdot {A(s)}}}}$

n_(i) and m_(i) are any positive integers.

FIG. 15 shows a structural block diagram of an improved all-harmonicdigital repetitive controller with a parallel structure. The transferfunction is as follows:

${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {\sum\limits_{i = 0}^{m}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot z^{\frac{N}{n}} \cdot {Q_{i}(z)}} - {Q_{i}^{2}(z)}}{z^{\frac{2N}{n}} - {2{{\cos( {\frac{2\pi}{n} \cdot i} )} \cdot z^{\frac{N}{n}} \cdot {Q_{i}(z)}}} - {Q_{i}^{2}(z)}} \cdot {A(z)}}}}$

n and m are both positive integers; when n is an even number, m=0, 1, .. . , n/2; and when n is an odd number, m=0, 1, . . . , [n/2].

FIG. 16 shows a structural block diagram of an improved any-harmonicdigital repetitive controller with a parallel structure. The transferfunction is as follows:

${G_{rc}(z)} = {\frac{c(z)}{e(z)} = {\sum\limits_{i = 0}^{l}{k_{i} \cdot \frac{{{\cos( {\frac{2\pi}{n_{1}} \cdot m_{i}} )} \cdot z^{\frac{N}{n_{i}}} \cdot {Q_{i}(z)}} - {Q_{i}^{2}(z)}}{z^{\frac{2N}{n_{i}}} - {2{{\cos( {\frac{2\pi}{n_{1}} \cdot m_{i}} )} \cdot z^{\frac{N}{n_{i}}} \cdot {Q_{i}(z)}}} - {Q_{i}^{2}(z)}} \cdot {A(z)}}}}$

n_(i) and m_(i) are any positive integers.

Example 2

The nk±m RC provided in Example 1 and the h-order nk±m-order-harmonicdigital repetitive controller obtained by further expansion may be addedto a general feedback control system in an insertion or cascade mannerfor canceling nk±m-order-harmonic components in control errors. Theaddition of the nk±m RC and the h-order nk±m-order-harmonic digitalrepetitive controller obtained by further expansion to the generalfeedback system in the insertion manner will be introduced belowrespectively:

(1) For nk±m RC

A specific embodiment of the nk±m RC provided by the disclosure isintroduced by taking the addition of the nk±m-order-harmonic digitalrepetitive controller to the general feedback system in the insertionmanner as an example in this example:

FIG. 17 shows a structural block diagram of adding an improvednk±m-order-harmonic digital repetitive controller to a general feedbackcontrol system. G_(rc)(z) is the improved nk±m-order-harmonic digitalrepetitive controller; G_(c)(z) is a conventional feedback controller;G_(p)(z) is a control object; y_(d)(z) is the reference input of thesystem and typically is the fundamental reference signal; y(z) is theactual output of the system; e(z) is the error of reference and actualsignals and is also the input signal of the repetitive controllerG_(rc)(z); c(z) is the output signal of the repetitive controllerG_(rc)(z) and is also added with the error signal e(z) to serve as theinput of the conventional feedback controller G_(c)(z); u(z) is theoutput signal of the conventional feedback controller G_(c)(z) and isalso the input signal of the control object G_(p)(z); and d(z) is thedisturbance input signal of the system, which is added to the outputsignal of the control object G_(p)(z) to form the actual output signaly(z).

FIG. 18 shows a structural block diagram of adding an improved h-ordernk±m-order-harmonic digital repetitive controller (h≥2) to a generalfeedback control system. A high-order nk±m-order-harmonic digitalrepetitive controller is also added to the general feedback controlsystem in the insertion manner.

In order to verify the effectiveness and practicability of the nk±m RCprovided by the disclosure, a simulation experiment based onMatlab/Simulink is carried out as follows:

In the case of a single-phase inverter with single-phase rectifierloads, since harmonics are mainly concentrated at (4k±1) (i.e. oddorders of 3, 5, 7, 9 and the like)-frequency components and it is oftennecessary to track a fundamental reference signal, it only needs to maken=4 and m=1, and error-free tracking of the fundamental reference signaland complete cancellation of odd-harmonics may be realized. A controltarget is to make the output voltage track the reference voltageaccurately, where the reference voltage is V_(ref)=156 sin 100πt. FIG.19A-FIG. 19D show steady-state output oscillograms and error convergencevariation diagrams under two complex controls of a conventionalrepetitive controller (CRC) (Zhou K, Wang D, “Digital repetitivelearning controller for three-phase CVCF PWM inverter[J]”, IEEETransactions on Industrial Electronics, 2001) and a 4k±1 RC provided bythe disclosure which are added respectively when t=0.1 s and therepetitive control gain k_(rc) is the same on the basis of an FC (statefeedback controller). FIG. 19A and FIG. 19B correspond to the CRC; FIG.19A is the steady-state output oscillogram, and FIG. 19B is the errorconvergence variation diagram. FIG. 19C and FIG. 19D correspond to the4k±1 RC provided by the disclosure; FIG. 19C is the steady-state outputoscillogram, and FIG. 19D is the error convergence variation diagram.

According to FIG. 19A-FIG. 19D, it can be seen that the total harmonicdistortions (THD) of the CRC and the 4k±1 RC are 0.57% and 1.1%,respectively, which may reach very small values, but with almost thesame harmonic suppression effect as the CRC, the CRC needs about 0.7 sto reach a steady state. However, the 4k±1 RC corresponding to thedisclosure only needs about 0.35 s to reach a steady state, andtherefore the error convergence speed of the 4k±1 RC is about twice thatof the CRC, i.e. the error convergence speed of the 4k±1 RCcorresponding to the disclosure is obviously higher.

In the case of a three-phase inverter with three-phase rectifier loads,since harmonics are mainly concentrated at (6k±1) (i.e. orders of 5, 7,11, 13 and the like)-order-harmonic frequency components and it is oftennecessary to track a fundamental reference signal, it only needs to maken=6 and m=1, and error-free tracking of the fundamental reference signaland complete cancellation of (6k±1)-order-harmonics may be realized. Acontrol target is to make the output voltage track the reference voltageaccurately, where the reference voltages are V_(abref)=220 sin 100πt,V_(bcref)=220 sin(100πt−⅔π), and V_(caref)=220 sin(100πt+⅔π). FIG.20A-FIG. 20D show steady-state output oscillograms and error convergencevariation diagrams under two complex controls of a CRC and a 6k±1 RCprovided by the disclosure which are added respectively when t=0.1 s andthe repetitive control gain k_(rc) is the same on the basis of an FC(state feedback controller). FIG. 20A and FIG. 20B correspond to theCRC; FIG. 20A is the steady-state output oscillogram, and FIG. 20B isthe error convergence variation diagram. FIG. 20C and FIG. 20Dcorrespond to the 6k±1 RC provided by the disclosure; FIG. 20C is thesteady-state output oscillogram, and FIG. 20D is the error convergencevariation diagram.

According to FIG. 20A-FIG. 20D, it can be seen that the total harmonicdistortions (THD) of the CRC and the 6k±1 RC are 0.38% and 1.05%,respectively, which may reach very small values, but with almost thesame harmonic suppression effect as the CRC, the CRC needs about 0.75 sto reach a steady state. However, the 6k±1 RC corresponding to thedisclosure only needs about 0.25 s to reach a steady state, andtherefore the error convergence speed of the 6k±1 RC is about triplethat of the CRC, i.e. the error convergence speed of the 6k±1 RCcorresponding to the disclosure is obviously higher.

(2) For h-Order Nk±m RC

The nk±m RC provided by the disclosure may be further expanded into anh-order nk±m-order-harmonic digital repetitive controller (h≥2). Here, a6k±1 RC and a second-order 6k±1 RC are taken as examples. Specificsimulation examples verify that compared with a first-ordernk±m-order-harmonic digital repetitive controller, the performance ofthe high-order nk±m-order-harmonic digital repetitive controller isimproved, the mismatch between a controller internal model and aperiodic signal to be tracked or canceled near a resonance frequencypoint may be suppressed to some extent, and the error convergence speedis higher.

FIG. 21A-FIG. 21D show error convergence variation diagrams and harmonicspectrograms under two complex controls of a 6k±1 RC and a providedsecond-order 6k±1 RC which are added respectively when t=0.1 s and therepetitive control gain k_(rc) is the same on the basis of an FC (statefeedback controller) with the frequency changed from 50 Hz to 49.8 Hz.FIG. 21A and FIG. 21B correspond to the 6k±1 RC; FIG. 21A is the errorconvergence variation diagram, and FIG. 21B is the harmonic spectrogram.FIG. 21C and FIG. 21D correspond to the second-order 6k±1 RC; FIG. 21Cis the error convergence variation diagram, and FIG. 21D is the harmonicspectrogram.

According to FIG. 21A-FIG. 21D, it can be seen that the errorconvergence time and the total harmonic distortion (THD) of the 6k±1 RCare 0.3 s and 1.59%, respectively, and the error convergence time andthe total harmonic distortion (THD) of the second-order 6k±1 RC are 0.2s and 1.42%, respectively. Therefore, when the mismatch occurs between acontroller internal model and a periodic signal to be tracked orcanceled near a resonance frequency point, the second-order 6k±1 RC hasa higher error convergence speed and a lower total harmonic distortion.

Example 3

The nk±m-order-harmonic digital repetitive controller provided inExample 1, the h-order nk±m-order-harmonic digital repetitive controllerobtained by further expansion, and the all/any-harmonic digitalrepetitive controller with a parallel structure may be added to ageneral feedback control system in an insertion or cascade manner forcanceling nk±m-order-harmonic components in control errors. The specificembodiment and the simulation experiment in which the nk±m RC and theh-order nk±m-order-harmonic digital repetitive controller are added tothe general feedback control system in the insertion manner are providedin Example 2. This example introduces a specific embodiment of adding tothe general feedback control system in the cascade manner.

FIG. 22 shows a structural block diagram of adding an improvednk±m-order-harmonic digital repetitive controller or a repetitivecontroller with a parallel structure or an h-order nk±m-order-harmonicdigital repetitive controller to a general feedback control system in acascade manner. G_(rc)(z) is the improved nk±m-order-harmonic digitalrepetitive controller; G_(c)(z) is a conventional feedback controller;G_(p)(z) is a control object; y_(d)(z) is the reference input of thesystem and typically is the fundamental reference signal; y(z) is theactual output of the system; e(z) is the error of reference and actualsignals and is also the input signal of the repetitive controllerG_(rc)(z); c(z) is the output signal of the repetitive controllerG_(rc)(z) and is also added with the error signal e(z) to serve as theinput of the conventional feedback controller G_(c)(z); u(z) is theoutput signal of the conventional feedback controller G_(c)(z) and isalso the input signal of the control object G_(p)(z); and d(z) is thedisturbance input signal of the system, which is added to the outputsignal of the control object G_(p)(z) to form the actual output signaly(z).

Some steps in the examples of the disclosure may be implemented throughsoftware, and corresponding software programs may be stored in areadable storage medium, such as an optical disk or a hard disk.

The foregoing descriptions are merely preferred examples of thedisclosure, but are not intended to limit the disclosure. Anymodification, equivalent substitution, improvement and the like madewithin the spirit and principle of the disclosure shall fall within theprotection scope of the disclosure.

What is claimed is:
 1. A repetitive controller, comprising: a repetitivecontrol gain module, a positive feedforward gain module, a subtractionloop, a first addition loop, a second addition loop, a first time-delaymodule, a second time-delay module and a third time-delay module;wherein the first time-delay module, the second time-delay module andthe third time-delay module are identical; wherein: an input end of therepetitive control gain module receives an input into the repetitivecontroller, an output end of the repetitive control gain moduleconnected to a first input end of the first addition loop, an output endof the first addition loop is connected to a first input end of thesecond addition loop, an output end of the second addition loop isconnected to an input end of the positive feedforward gain module and aninput end of the first time-delay module, an output end of the positivefeedforward gain module is connected to a positive input end of thesubtraction loop, an output end of the first time-delay module isconnected to a negative input end of the subtraction loop, an output endof the subtraction loop is connected to an input end of the secondtime-delay module, an output end of the second time-delay module isconnected to a second input end of the first addition loop, an outputfrom the repetitive controller is received from the output end of thesecond time-delay module, the output end of the positive feedforwardgain module is also connected to an input end of the third time-delaymodule, and an output end of the third time-delay module is connected toa second input end of the second addition loop.
 2. The repetitivecontroller according to claim 1, further comprising: a first low passfilter, a second low pass filter, a third low pass filter, and a phaselead compensation module; wherein the output end of the first time-delaymodule is connected to the negative input end of the subtraction loopthrough the first low pass filter, the output end of the secondtime-delay module is connected to the second input end of the firstaddition loop through the second low pass filter, the output from therepetitive controller is received from the output end of the secondtime-delay module through the second low pass filter and then throughthe phase lead compensation module, the output end of the positivefeedforward gain module is connected to the input end of the thirdtime-delay module through the third low pass filter.
 3. A method ofcontrolling the repetitive controller of claim 1, comprising: using therepetitive control gain module to perform repetitive control gain on aninput quantity of the repetitive controller to obtain an output quantityof the repetitive control gain module; using the positive feedforwardgain module to perform positive feedforward gain on an output quantityof the second addition loop to obtain an output quantity of the positivefeedforward gain module; using the first addition loop to add the outputquantity of the repetitive control gain module and an output quantity ofthe subtraction loop output delayed by the second time-delay module in adelay manner to obtain an output quantity of the first addition loop;using the second addition loop to add the output quantity of the firstaddition loop and the output quantity of the positive feedforward gainmodule output delayed by the third time-delay module to obtain an outputquantity of the second addition loop; using the subtraction loop tosubtract the output quantity of the positive feedforward gain modulefrom the output quantity of the second addition loop output delayed bythe first time-delay module to obtain the output quantity of thesubtraction loop; using the first time-delay module to delay the outputquantity of the second addition loop; using the second time-delay moduleto delay the output quantity of the subtraction loop; and using thethird time-delay module to delay the output quantity of the positivefeedforward gain module.
 4. The method according to claim 3, furthercomprising: adjusting the repetitive control gain to adjust a speed ofthe repetitive controller to track or cancel specific harmonics.
 5. Themethod according to claim 3, further comprising: determining parametersof the positive feedforward gain module according to an order ofharmonics to be tracked or canceled.
 6. The method according to claim 3,wherein the first, second or third time-delay module is an analog ordigital time-delay module, and a transfer function of the repetitivecontroller is:${G_{rc}(s)} = {\frac{c(s)}{e(s)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi\; m}{n} )} \cdot e^{\frac{{sT}_{0}}{n}}} - 1}{e^{\frac{2{sT}_{0}}{n}} - {2{{\cos( \frac{2\pi\; m}{n} )} \cdot e^{\frac{{sT}_{0}}{n}}}} + 1}}}$${{or}\mspace{14mu}{G_{rc}(z)}} = {\frac{c(z)}{e(z)} = {k_{rc} \cdot \frac{{{\cos( \frac{2\pi\; m}{n} )} \cdot z^{\frac{N}{n}}} - 1}{z^{\frac{2N}{n}} - {2{{\cos( \frac{2\pi\; m}{n} )} \cdot z^{\frac{N}{n}}}} + 1}}}$wherein c( )is the output quantity of the repetitive controller, e( )isthe input quantity of the repetitive controller, k_(rc) is a repetitivecontrol gain parameter, s is a Laplace variable of a continuous timesystem, and z is a variable of z transformation of a discrete timesystem; N=T₀/T_(s), is an integer, T₀ is a fundamental period,T₀=2π/ω₀=l/f₀, f₀ is a fundamental frequency, ω₀ is a fundamentalangular frequency, T_(s) is a sampling period, n, k and m are integersnot less than zero, n≠0, and n>m.
 7. A method of using the repetitivecontroller of claim 1, for canceling nk±m-order-harmonic components incontrol errors in a feedback control system, wherein a transfer functionof the repetitive controller is:${G_{rc}(z)} = {{\frac{k_{rc}}{2}\lbrack {\frac{e^{j\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}{1 - {e^{j\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}} + \frac{e^{{- j}\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}{1 - {e^{{- j}\; 2\;\pi\;{m/n}} \cdot z^{{- N}/n} \cdot {Q(z)}}}} \rbrack} \cdot {A(z)}}$wherein Q(z) is a low pass filter, and A(z) is a phase lead compensationmodule; and the feedback control system is stable when the following twoconditions are met: (1) poles of a transfer function of a closed-loopsystem before the repetitive controller is inserted are located in aunit circle; and (2) a repetitive control gain parameter k, in theinserted controller meets 0<k_(rc)<2.